Nonequilibrium properties of many-body statistical systems on macroscopic
scales are usually well described by hydrodynamic equations.
This is justified by the fact that macroscopic fluctuations of conserved quantities
(and order parameters close to criticality)
must evolve slowly with respect to microscopic time-scales
to reach a steady state. If the relevant correlations have long-time tails,
standard diffusive behavior may break down as it happens generically in
nonlinear, low-dimensional systems [1]}, [2]}, [3]}, [4]}.
Anomalous transport in such many-body systems can be effectively described
by a random Lévy walk [5]} of the energy carriers, as demonstrated extensively in the literature
[6]}, [7]}, [8]}. This leads naturally to
consider hydrodynamic equations where the standard Laplacian operator is replaced by
a fractional one [9]}, [10]}, [11]}, [12]}.
Nonlinear coupling among hydrodynamic fields yields slow algebraic decay of current correlations at equilibrium and effective non-local equations.
A theoretical justification of such a behavior has been obtained by the nonlinear fluctuating hydrodynamics approach, whereby long-wavelength fluctuations are described in terms of the Kardar-Parisi-Zhang equations [13]}.
Although the resulting predictions are well confirmed by numerical
simulations [14]}, [15]}, [16]}, there may be significant scale-effects, especially
when the dynamics is only weakly chaotic [17]}.
